Legendre polynomials

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematically beautiful properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

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In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x)=1{\displaystyle w(x)=1} over the interval [−1,1]{\displaystyle [-1,1]}. That is, Pn(x){\displaystyle P_{n}(x)} is a polynomial of degree n{\displaystyle n}, such that

This determines the polynomials completely up to an overall scale factor, which is fixed by the standardization
Pn(1)=1{\displaystyle P_{n}(1)=1}. That this is a constructive definition is seen thus: P0(x)=1{\displaystyle P_{0}(x)=1} is the only correctly standardized polynomial of degree 0. P1(x){\displaystyle P_{1}(x)} must be orthogonal to P0{\displaystyle P_{0}}, leading to P1(x)=x{\displaystyle P_{1}(x)=x}, and P2(x){\displaystyle P_{2}(x)} is determined by demanding orthogonality to P0{\displaystyle P_{0}} and P1{\displaystyle P_{1}}, and so on. Pn{\displaystyle P_{n}} is fixed by demanding orthogonality to all Pm{\displaystyle P_{m}} with m<n{\displaystyle m<n}. This gives n{\displaystyle n} conditions, which, along with the standardization Pn(1)=1{\displaystyle P_{n}(1)=1} fixes all n+1{\displaystyle n+1} coefficients in Pn(x){\displaystyle P_{n}(x)}. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of x{\displaystyle x} given below.

This definition of the Pn{\displaystyle P_{n}}'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, x,x2,x3,…{\displaystyle x,x^{2},x^{3},\ldots }. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line [0,∞){\displaystyle [0,\infty )}, and the Hermite polynomials, orthogonal over the full line (−∞,∞){\displaystyle (-\infty ,\infty )}, with weight functions that are the most natural analytic functions that ensure convergence of all integrals.

The coefficient of tn{\displaystyle t^{n}} is a polynomial in x{\displaystyle x} of degree n{\displaystyle n}. Expanding up to t1{\displaystyle t^{1}} gives

P0(x)=1,P1(x)=x.{\displaystyle P_{0}(x)=1\,,\quad P_{1}(x)=x.}

Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.

It is possible to obtain the higher Pn{\displaystyle P_{n}}'s without resorting to direct expansion of the Taylor series, however. Eq. 2 is differentiated with respect to t on both sides and rearranged to obtain

This differential equation has regular singular points at x = ±1 so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for |x| < 1 in general. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of Sturm–Liouville theory. We rewrite the differential equation as an eigenvalue problem,

with the eigenvalue λ{\displaystyle \lambda } in lieu of n(n+1){\displaystyle n(n+1)}. If we demand that the solution be regular at
x=±1{\displaystyle x=\pm 1}, the differential operator on the left is Hermitean. The eigenvalues are found to be of the form
n(n + 1), with n=0,1,2,…{\displaystyle n=0,1,2,\ldots }, and the eigenfunctions are the Pn(x){\displaystyle P_{n}(x)}. The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory.

In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as Pn(cos⁡θ){\displaystyle P_{n}(\cos \theta )} where θ{\displaystyle \theta } is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning.

The standardization Pn(1)=1{\displaystyle P_{n}(1)=1} fixes the normalization of the Legendre polynomials
(with respect to the L2 norm on the interval −1 ≤ x ≤ 1). Since they are also orthogonal with respect to the same norm, the two statements can be combined into the single equation,

where the last, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.[clarification needed]

where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of Laplace's equation of the static potential, ∇2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where ẑ is the axis of symmetry and θ is the angle between the position of the observer and the ẑ axis (the zenith angle), the solution for the potential will be

where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.

Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion.

The trigonometric functions cos nθ, also denoted as the Chebyshev polynomialsTn(cos θ) ≡ cos nθ, can also be multipole expanded by the Legendre polynomials Pn(cos θ). The first several orders are as follows:

which follows from considering the orthogonality relation with P0(x)=1{\displaystyle P_{0}(x)=1}. It is convenient when a Legendre series ∑iaiPi{\displaystyle \sum _{i}a_{i}P_{i}} is used to approximate a function or experimental data: the average of the series over the interval [−1, 1] is simply given by the leading expansion coefficient a0{\displaystyle a_{0}}.

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that

All n{\displaystyle n} zeros of Pn(x){\displaystyle P_{n}(x)} are real, distinct from each other, and lie in the interval (−1,1){\displaystyle (-1,1)}. Further, if we regard them as dividing the interval [−1,1]{\displaystyle [-1,1]} into n+1{\displaystyle n+1} subintervals, each subinterval will contain exactly one zero of Pn+1{\displaystyle P_{n+1}}. This is known as the interlacing property. Because of the parity property it is evident that if xk{\displaystyle x_{k}} is a zero of Pn(x){\displaystyle P_{n}(x)}, so is −xk{\displaystyle -x_{k}}. These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the Pn{\displaystyle P_{n}}'s is known as Gauss-Legendre quadrature.